A social decision function (SDF) à la Sen (1970) is defined as a collective choice rule whose range is restricted to social preference relations which generate a choice function. From Gaertner (2009), a preference relation $R$ generates a choice function over a set $X$ if and only if $R$ is reflexive, complete and acyclical over $X$. I thus struggle to understand where exactly the difference between a SDF and a social choice function (SCF) à la Gibbard-Satterthwaite lies. A SCF is itself a choice function, so the preference relation $R$ generating it must satisfy the same conditions as the one generating a SDF.
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$\begingroup$ I don't think there can be a single correct answer to this question. Definitions vary across writers (and time). The onus is on the writer in a particular piece to precisely define what she means by any term. $\endgroup$– user18Sep 5, 2020 at 2:47
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Let $X$ be the set of alternatives.
A social decision function maps profiles of preference orderings to relations on $X$ such that every nonempty subset of $X$ has at least one maximum under this relation.
A social choice function maps profiles of preference orderings to elements of $X$.
Now let $P$ be a profile of preferences, $f$ a social decision function, and $g$ a social choice function.
There might be more than one $f(P)$-maximum in $X$, so $f$, in contrast to $g$, will not always pin down a single choice in $X$.
On the other hand, suppose the alternative $g(P)$ is not available for some reason (say, the winning candidate died). Then $g$ is of no help in finding an alternative from the remaining set of alternatives $X\setminus\{g(P)\}$. But $f(P)$ will allow us to rank the alternatives in this remaining set (provided $g(P)$ is not the only element of $X$), though, again, there might be more than on $f(P)$-maximum in this set.
answered Sep 4, 2020 at 20:53
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$\begingroup$ Thank you! So in other words, the $R$ generating a social decision function is complete over all of $X$, while the one generating a social choice function must only be complete for the best alternative? What, then, is the difference between a SDF and a social welfare function? Can't we construct a social welfare function from the information encoded in a SDF (assuming acyclicality?) $\endgroup$ Sep 5, 2020 at 8:59
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$\begingroup$ Edit to my comment above: is it correct that the only difference between SWFs and SDFs is that the former require transitivity ($xPy$ and $yIz \Rightarrow xPz$) while the latter only quasi-transitivity (e.g. it could be that $xPy$ and $yIz$ and $xIz$, such as in Sen's (1969) Pareto extension rule)? $\endgroup$ Sep 5, 2020 at 11:55
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$\begingroup$ In Sen's 1970 book, the values of an (Arrovian) SWF have to be "orderings" (reflexive, complete, and transitive relations) on $X$, while the values of an SDF need only induce a choice function for all nonempty subsets. I don't have a characterization of that property at hand, but for infinite sets, an SWF need not be an SDF. $\endgroup$ Sep 5, 2020 at 12:47